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 2006-09-20, 12:26 #1 AntonVrba     Jun 2005 2·72 Posts Sum of all integer digits of all primes between 1 an n With the discovery of M44 (congratulations GIMPS) I pondered over the thought if it is possible to count/calculate all the ones of all the primes between 2 and M44. It is a bit off-topic but the result is interesting. To calculate or estimate the number of ones, I set about as follows. Define $c_b(n)$ as the sum of all base-b integer digits between 1 and n and can be expressed as. $c_b(n)=\sum_{k = 0}^{m}\,\,{d_k}\,\left( b^k\, \frac{{d_k} + k\, \left( b - 1\right) - 1}{2} + {mod (n,\,{b^k})} + 1\right)$ Above has the spot values $c_b\left( b^m-1 \right) =\frac{m\left( b-1\right)b^m}{2}$ Now assume $m$ to be large then $m=log_b(b^m-1)=ln(b^m-1)/ln(b)$ and proportioning $c_b(b^m-1)$ to the number of primes between $1$ and $b^m-1$ which is approximated in the Prime Number Theorem as $(b^m-1)/ln(b^m-1)$ we obtain the unexpected result that the cumulative sum of all base-b integer digits of all the primes between $1$ and prime $b^m-1$ approximates to $\frac{(b-1)(b^m-1)}{2\, ln(b)}\,$ Conjecture The ratio$\kappa_b$ defined as "the sum of all base-b digits of all the primes between 1 and n" to "n", converges to the constant $\frac{b-1}{2\, ln(b)$ for increasing n. $\kappa_2=0.7213476$ and $\kappa_{10}=1.9543251$ A computation check confirms above tendency already at relatively small values of n. Is above already known or have I introduced a new constant? In parctice how will the constant depart from above definition? Regards Anton Vrba
 2006-09-20, 12:34 #2 AntonVrba     Jun 2005 1428 Posts For the Mathematica users Code: CumSumDigits[n_, b_] := Module[ {ss, p, d0, d1, m} , ss = p = d0 = 0 ; m = n ; While[m > 0, { If[(d1 = Mod[m, b]) != 0, { ss += d1( b^p(b - 1) p + b^p (d1 - 1) + 2(d0 + 1))/2, d0 += b^p d1}], p += 1, m = IntegerPart[m/b], }] ; Return [ss ] ; ] and can be checked by evaluating Code: CumSumDigits[(10^123 - 1)/9, 10] - CumSumDigits[(10^123 - 1)/9 - 1, 10]
 2006-09-20, 17:20 #3 AntonVrba     Jun 2005 2×72 Posts here is a base-10 evaluation of $\kappa_{10}$ counting up to Code: Prime Actual Calculated Actual/Calculated 99999989 2.09217 2.07412 1.0087 999999937 2.07641 2.05933 1.0083 9999999967 2.06389 2.00702 1.0078 99999999977 2.05366 2.03844 1.00747 Actual cumulative digit count of all primes 2 to p divided by the last prime p (C program) Calculated value $\frac{\pi(p)\, c_{10}(p)}{p^2}$ $\pi(p)$ being the prime counting function An on the same basis a base-2 evaluation for $\kappa_2$ Code: 32749 0.845614 0.804225 1.05146 65521 0.837533 0.798769 1.04853 131071 0.832198 0.794553 1.04738 262139 0.825139 0.789684 1.0449 524287 0.820511 0.78624 1.04359 1048573 0.815104 0.782262 1.04198 2097143 0.810823 0.779118 1.04069 4194301 0.806539 0.776155 1.03915 8388593 0.802665 0.773417 1.03782 16777213 0.798989 0.770954 1.03636 33554393 0.795861 0.768785 1.03522 67108859 0.792815 0.766688 1.03408 134217689 0.790022 0.764787 1.033 268435399 0.787444 0.763058 1.03196 536870909 0.785043 0.76144 1.031 1073741789 0.782845 0.75996 1.03011 2147483647 0.780761 0.758568 1.02926 Last fiddled with by AntonVrba on 2006-09-20 at 18:11

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